All Questions ⁺⁰235719 Questions

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Geometry

In triangle $PQR,$ $M$ is the midpoint of $\overline{PQ}.$ Let $X$ be the point on $\overline{QR}$ such that $\overline{PX}$ bisects $\angle QPR,$ and let the perpendicular bisector of $\overline{PQ}$ intersect $\overline{PX}$ at $Y.$ If PQ = 36, find the read more ..

eramsby1O1O Sep 21, 2024

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5

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+1839

Algebra

The system of equations \frac{xy}{x + y} = 1, \quad \frac{xz}{x + z} = 1, \quad \frac{yz}{y + z} = 0 has exactly one solution. What is $z$ in this solution?

ABJeIIy Sep 21, 2024

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+834

Algebra

Will and Grace are canoeing on a lake. Will rows at $50$ meters per minute and Grace rows at $30$ meters per minute. Will starts rowing at $2$ p.m. from the west end of the lake, and Grace starts rowing from the east end of the lake at $2{:}45$ p.m. read more ..

gnistory Sep 21, 2024

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Algebra

I have to paint one side of a wall. The wall is $15$ meters tall and $60$ meters long. Each gallon of paint covers $400$ square feet. If a foot is approximately $0.3048$ meters, then what is the smallest whole number of gallons I can buy and read more ..

gnistory Sep 21, 2024

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Algebra

I drove to the beach at a rate of $40$ miles per hour. If I had driven at a rate of $55$ miles per hour instead, then I would have arrived $25$ minutes earlier. How many miles did I drive?

gnistory Sep 21, 2024

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+306

Counting

(a) Let $ABC$ be an equilateral triangle, centered at $O.$ A point $P$ is chosen at random inside the triangle. Find the probability that $P$ is closer to $O$ than to any of the vertices. (In other words, find the probability that $OP$ is shorter than $AP,$ read more ..

nathanl6656 Sep 21, 2024

Sep 20, 2024

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Counting

Triangle XYZ is equilateral, with O as its center. A point P is chosen at random. Find the probability that P is closer to point O than to any of the side lengths.

nathanl6656 Sep 20, 2024

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+306

Counting

The entries in a certain row of Pascal's triangle are
\[1, n, \dots, n, 1.\]
The average of the entries in this row is 2. Find $n$.

nathanl6656 Sep 20, 2024

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5

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+306

Counting

In a row of five squares, each square is to be colored either red, yellow, or blue, so that no two consecutive squares have the same color. How many ways are there to color the five squares, if there must be at least three yellow squares?

nathanl6656 Sep 20, 2024

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+829

Algebra

Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

AnswerscorrectIy Sep 20, 2024

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+829

Algebra

Solve the system of equations
y = \log_2 (2x)
y = log_4 (16 + x)

AnswerscorrectIy Sep 20, 2024

Sep 19, 2024

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+1839

Algebra

Find one ordered pair $(x,y)$ of real numbers such that $x + y = 10$ and $x^3 + y^3 = 162 + x^2 + y^2.$

ABJeIIy Sep 19, 2024

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9

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+1839

Algebra

Find the minimum value of 9^x - 2 \cdot 3^x + 1 over all real numbers x.

ABJeIIy Sep 19, 2024

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+1839

Algebra

I own a large truck, and my neighbor owns four small trucks that are all identical. My truck can carry a load of at least 600 pounds more than each of her trucks, but no more than 1/2 of the total load her four trucks combined can carry. Based on these read more ..

ABJeIIy Sep 19, 2024

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+829

Geometry

A semicircle is inscribed in triangle XYZ so that its diameter lies on YZ, and is tangent to the other two sides. If XY = 10, XZ = 10, and YZ = 10*sqrt(2), then find the area of the semicircle.

AnswerscorrectIy Sep 19, 2024

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+829

Geometry

Two circles are externally tangent at T. The line AB is a common external tangent to the two circles, and P is the foot of the altitude from T to line AB. Find the length AB.

AnswerscorrectIy Sep 19, 2024

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Algebra

The equation of the line below can be expressed in the form
\frac{x}{a} + \frac{y}{b} = 1.
Find a. read more ..

onyuIee Sep 19, 2024

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